I - In the previous two posts we discussed MSK performance in an AWGN channel, first presenting the MATLAB/OCTAVE Code for one sample per symbol case [Post 1], and then extending it to the more general case of multiple samples per symbol [Post 2]. This helps us visualize the underlying beauty of Continuous Phase Modulation (CPM) which reduces out of band energy and consequently lowers Adjacent Channel Interference (ACI). We also briefly touched upon the case of MSK in Rayleigh fading, but did not go into the details. So here we take a deeper dive.

II - On the face of it its quite simple. Just as we add an Independent and Identically distributed (IID) Complex Gaussian Random Variable to the signal we can do the same for the fading channel (the only difference being that the channel is multiplicative and noise is additive). So the received signal would look something like this y=h.*s+n where 'h' and 'n' are defined as given below ('Ns' is the samples/symbol and 'Nbits' is the total number of bits or symbols):

n=sigma*(randn(1,Ns*Nbits)+i*randn(1,Ns*Nbits))
h=(1/sqrt(2))*(randn(1,Ns*Nbits)+i*randn(1,Ns*Nbits)) 

III - But there is a caveat. If lets say we have four samples per symbol and we multiple the signal vector 's' with channel coefficient vector 'h', with 'h' having four IID values then we are providing a diversity advantage and we will be experiencing a gain in the performance like we do in Maximal Ratio Combining (assuming we multiply the received signal with the conjugate of the channel before correlation with the basis function). Indeed this is validated through Monte Carlo simulation where the BER continuously improves as the number of samples per symbol increases.

IV - So what is the solution? One solution is to assume that the channel is Quasi Static i.e. it is constant over the duration of a single symbol and then changes in the next symbol (without knowledge of prior value or in other words without memory). In our simulation we considered number of samples per symbol to be four and the channel to be static over 100 symbols (or bits assuming 1 bit/symbol). In our simulations we found the simulated BER to exactly match the theoretical BER in Rayleigh channel which is given as 0.5*(1-sqrt(EbNo/(EbNo+1))).

OCTAVE Code for MSK in Rayleigh Fading

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       MINIMUM SHIFT KEYING      %
%     GENERALIZED MATRIX BASED    %
%  BER OF MSK IN RAYLEIGH FADING  %
%         www.raymaps.com         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all

Ns=4;
Nbits=1e6;
Nstat=100;
Eb=Ns;
EbNodB=10;

% MODULATION
bits_in=round(rand(1,Nbits));
differential_bits=abs(diff([0, bits_in]));
bipolar_symbols=differential_bits*2-1;
symbols_oversampled=(ones(Ns,1))*bipolar_symbols;
symbols_oversampled=(1/Ns)*transpose(symbols_oversampled(:));
cumulative_phase=(pi/2)*cumsum(symbols_oversampled);
tx_signal=exp(i*(cumulative_phase));

% CHANNEL (NOISE AND FADING)
EbNo=10^(EbNodB/10);
sigma=sqrt(Eb/(2*EbNo));
AWGN_noise=randn(1,Ns*Nbits)+i*randn(1,Ns*Nbits);
ray_chan=(1/sqrt(2))*(randn(1,Nbits/Nstat)+i*randn(1,Nbits/Nstat));
ray_chan_mat=ones(Nstat*Ns,1)*ray_chan;
ray_chan_vec=transpose(ray_chan_mat(:));
rx_signal=ray_chan_vec.*tx_signal+sigma*AWGN_noise;

% DEMODULATION (EQUALIZATION, CORRELATION, DECISION)
rx_signal=rx_signal.*conj(ray_chan_vec);

sin_waveform=sin(pi/(2*Ns):pi/(2*Ns):pi);
rx_real=[real(rx_signal(Ns+1:end)), zeros(1,Ns)];
rx_imag=[imag(rx_signal(1:end))];
rx_I_metric=sin_waveform*reshape(rx_real, 2*Ns, Nbits/2);
rx_Q_metric=sin_waveform*reshape(rx_imag, 2*Ns, Nbits/2);

n=1:Nbits/2;
rx_I_metric=(-1).^(n+1).*rx_I_metric;
rx_Q_metric=(-1).^(n+1).*rx_Q_metric;
rx_combined=[rx_Q_metric; rx_I_metric];
rx_combined=transpose(rx_combined(:));
demodulated_symbols=sign(rx_combined);
bits_out=demodulated_symbols*0.5+0.5;

%BER CALCULATION
ber_theoretical=0.5*(1-sqrt(EbNo/(EbNo+1)))
ber_simulated=sum(bits_out!=bits_in)/Nbits

Simulation Results

Note:

1. The code above also works for Ns=1 i.e. for one sample per symbol. Which as discussed earlier is just like doing BPSK simulation and does not require the channel to be Quasi Static.
2. It must also be noted that same results can be obtained by using a different method of equalization, division by the channel instead of multiplication with the complex conjugate.
3. Please note that we perform differential encoding before modulation at the transmitter side. This results in a very elegant coherent receiver architecture with BER performance same as BPSK [1].

[1] Digital Communication over Fading Channels by Marvin K. Simon, Mohamed-Slim Alouini